VERY EASY, WILL GIVE 50 POINTS FOR CORRECT ANSWER ASAP AND WILL GIVE BRAINLIEST.

Answer:

495

Step-by-step explanation:

using the definition

n[tex]C_{r}[/tex] = [tex]\frac{n!}{r!(n-r)!}[/tex]

where n! = n(n - 1)(n - 2) ... × 3 × 2 × 1

then

12[tex]C_{4}[/tex]

= [tex]\frac{12!}{4!(12-4)!}[/tex]

= [tex]\frac{12!}{4!(8!)}[/tex]

cancel 8! on numerator/ denominator

= [tex]\frac{12(11)(10)(9)}{4!}[/tex]

= [tex]\frac{11880}{4(3)(2)(1)}[/tex]

= [tex]\frac{11880}{24}[/tex]

= 495

Answer:  j(x) = (x-1)(x+2)

The x-1 part is because of the root x = 1

The root x = -2 leads to the factor x+2

Answer:

100

Step-by-step explanation:

#20 naira - 20 * 100

= 2000kobo

2000/20

100

QED✅✅

you can support by rating brainly it's very much appreciated ✅ ✅

Markdowns are usually applied before any additional discounts or promotions. In this case, the markdown reduced the regular price by 33%, resulting in the final price of $26.80.

Jules Girard bought 1 gallon of paint at a 33% markdown from its regular price of $39.99. To calculate the final price after the markdown, we can apply the percentage decrease to the regular price.

The markdown of 33% means that the price is reduced by 33% of the regular price. To calculate the amount of the markdown, we can multiply the regular price by the percentage decrease:

Markdown amount = 33% of $39.99 = 0.33 * $39.99

Next, we can subtract the markdown amount from the regular price to find the final price:

Final price = Regular price - Markdown amount = $39.99 - (0.33 * $39.99)

Calculating the expression within parentheses:

Final price = $39.99 - $13.19

Final price = $26.80

Therefore, Jules Girard bought 1 gallon of paint at a 33% markdown, resulting in a final price of $26.80.

Markdowns are usually applied before any additional discounts or promotions. In this case, the markdown reduced the regular price by 33%, resulting in the final price of $26.80.

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Step-by-step explanation:

To find the zeros of the polynomial p(x) = (2x^2 - 9x + 7)(x - 2), we set p(x) equal to zero and solve for x:

(2x^2 - 9x + 7)(x - 2) = 0

This equation will be satisfied if either of the factors on the left side equals zero. So we set each factor equal to zero and solve for x separately.

Setting 2x^2 - 9x + 7 = 0:

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)

For this equation, a = 2, b = -9, and c = 7.

Plugging these values into the quadratic formula, we have:

x = (-(-9) ± √((-9)^2 - 4 * 2 * 7)) / (2 * 2)

x = (9 ± √(81 - 56)) / 4

x = (9 ± √25) / 4

x = (9 ± 5) / 4

So we have two solutions:

x1 = (9 + 5) / 4 = 14 / 4 = 3.5

x2 = (9 - 5) / 4 = 4 / 4 = 1

Setting x - 2 = 0:

x = 2

Therefore, the zeros of the polynomial p(x) = (2x^2 - 9x + 7)(x - 2) are x = 3.5, 1, and 2.

first off, let's take a peek at the picture above

hmmm the hyperbola is opening sideways, that means it has a horizontal traverse axis, it also means that the positive fraction will be the one with the "x" variable in it.

now, the length of the horizontal traverse axis is 4 units, from vertex to vertex, that means the "a" component of the hyperbola is half that or 2 units, and 2² = 4, with a center at the origin.

[tex]\textit{hyperbolas, horizontal traverse axis } \\\\ \cfrac{(x- h)^2}{ a^2}-\cfrac{(y- k)^2}{ b^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h\pm a, k)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{ a ^2 + b ^2} \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{(x- 0)^2}{ 2^2}-\cfrac{(y- 0)^2}{ (\sqrt{3})^2}=1\implies {\Large \begin{array}{llll} \cfrac{x^2}{4}-\cfrac{y^2}{3}=1 \end{array}}[/tex]

When viewed through the magnifying glass, the termite appears to be approximately 1.58 mm in length.

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Mikey Johnson shipped out 34 2/7 pounds of electrical supplies. The supplies are placed in individual packets that weigh 2 1/7 pounds each. Therefore, Mikey shipped out 16 packets of electrical supplies.

To solve the problem, we can use the following steps.Step 1: Find the weight of each packet.

We are given that the weight of each packet is 2 1/7 pounds.

To convert this mixed number into an improper fraction, we can multiply the whole number by the denominator and add the numerator.

This gives us: 2 1/7 = (2 × 7 + 1) / 7= 15 / 7 pounds.

Therefore, the weight of each packet is 15/7 pounds.

Now, divide the total weight by the weight of each packet.

We are given that the total weight of the supplies shipped out is 34 2/7 pounds.

To convert this mixed number into an improper fraction, we can multiply the whole number by the denominator and add the numerator.

This gives us: 34 2/7 = (34 × 7 + 2) / 7= 240 / 7 pounds.

Therefore, the total weight of the supplies is 240/7 pounds.

To find the number of packets that Mikey shipped out, we can divide the total weight by the weight of each packet.

This gives us: 240/7 ÷ 15/7 = 240/7 × 7/15= 16.

Therefore, Mikey shipped out 16 packets of electrical supplies.

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We subtract the cost of wine when buying one bottle per week from the cost of wine when buying two bottles per weekend. Miranda saves $780 in one year by buying only one bottle of wine per week instead of two.

To calculate how much Miranda saves in one year by buying only one bottle of wine per week instead of two, we can follow these steps:

Determine the number of bottles of wine she buys in a year:

Miranda used to buy 2 bottles of wine per weekend, so in a week, she bought 2 bottles.

Since there are 52 weeks in a year, the number of bottles she bought in a year is 2 bottles/week * 52 weeks = 104 bottles.

Calculate the total cost of wine for the year:

Since each bottle costs $15, the total cost of wine for the year when buying 2 bottles per week would be 104 bottles * $15/bottle = $1560.

Calculate the cost of wine for one year when buying only one bottle per week:

With Miranda's decision to buy one bottle per week, the total number of bottles she buys in a year is 1 bottle/week * 52 weeks = 52 bottles.

Therefore, the cost of wine for one year when buying only one bottle per week would be 52 bottles * $15/bottle = $780.

Calculate the amount saved in one year:

To determine the amount saved, we subtract the cost of wine when buying one bottle per week from the cost of wine when buying two bottles per weekend.

Amount saved = $1560 - $780 = $780.

Therefore, Miranda saves $780 in one year by buying only one bottle of wine per week instead of two.

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Mark's cost does not vary directly with time spent talking, it indicates the presence of other factors, such as pricing structure, additional services, peak hours, bundled packages, or promotions, that influence the overall cost. These factors can lead to non-linear cost variations and a lack of direct proportionality between the cost and time spent talking.

If Mark's cost does not vary directly with the time spent talking, it means that there is not a simple proportional relationship between the two variables. In other words, doubling the time spent talking does not necessarily result in doubling the cost. This indicates that there are other factors influencing Mark's cost besides just the time spent talking.

There can be several reasons why Mark's cost does not vary directly with time spent talking:

Pricing structure: Mark's cost may be determined by a pricing structure that includes fixed fees or additional charges based on certain conditions. For example, there might be a base rate for a certain duration of talk time, and any additional time may be billed at a different rate.

Additional services: Mark's cost might include additional services or features that are not solely dependent on talk time. For instance, there could be charges for international calling, call forwarding, or other value-added services.

Peak hours or special rates: Mark's cost might be affected by peak hours or special rates. Some service providers offer different pricing during certain times of the day or specific days of the week. This could result in non-linear cost variations with respect to time spent talking.

Bundled packages: Mark's cost might be part of a bundled package that includes other services like internet or television. In such cases, the overall cost may be influenced by factors other than talk time.

Promotions or discounts: Mark's cost may be subject to promotional offers or discounts that are not directly related to the time spent talking.

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The missing part indicated in the figures are ? = 12, TX = 9 and x = 20

Figure a

The missing part can be calculated using the following equation

?/(11 - 5) = 22/11

Evaluate the difference

?/6 = 22/11

So, we have

? = 6 * 22/11

Evaluate the expression

? = 12

Figure b

The missing part can be calculated using the following equation

TX/3 = 6/2

So, we have

TX = 3 * 6/2

Evaluate

TX = 9

Figure c

The value of x can be calculated using the following equation

1/4x + 6 = 2x - 29

So, we have

x + 24 = 8x - 116

Evaluate

-7x = -140

Divide

x = 20

Hence, the value of x is 20

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Answer:

x < 4.5

Step-by-step explanation:

To solve the inequality 24 > 6x - 3, we can follow these steps:

Start by adding 3 to both sides of the inequality to isolate the term with the variable:

24 + 3 > 6x - 3 + 3

This simplifies to:

27 > 6x

Divide both sides of the inequality by 6 to solve for x:

27/6 > 6x/6

Simplifying further:

4.5 > x

Therefore, the solution to the inequality 24 > 6x - 3 is x < 4.5.

Hope this helps!

The solution to the inequality is :

↬ [tex]\bf{x < \dfrac{9}{2}}[/tex]

Solution:

Let's solve [tex]\bf24\:\textgreater\:6x-3}[/tex].

First, add 3 to each side:

[tex]\sf{24+3\:\textgreater \:6x}[/tex]

[tex]\sf{27 > 6x}[/tex]

Divide each side by 6

[tex]\sf{27/6 > x}[/tex]

[tex]\sf{x < 27/6}[/tex]

[tex]\sf{x < 9/2}[/tex]

Based on the given functions, y = -5 and y = 3x + 0.5 are said to be affine functions. Hence option A and D are correct.

An affine function is seen as a function that can be shown as y = mx + b, where m and b are constants.

So looking at the functions given:

y = -5

y = 1 - 5x²

y = -2x²

y = 3x + 0.5

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Answer: I think it's H, if it isn't and you get a second try it should be N

Step-by-step explanation: Both H and N start with 0/-3 except F which means F is wrong. Maybe you should wait for a better answer...but it shouldn't be F

The length of BD of the given triangle is: 15

The perimeter of a triangle is simply defined as the length of the boundary of that triangle.

Now, we are given that:

AB = BC = a

Perimeter of triangle ABC = 50

Perimeter of triangle ABD = 40

Thus:

2a + 2b = 50

a + b = 25     ------(1)

Similarly:

a + b + h = 40

a + b = 40 - h   -----(2)

Put 40 - h for a + b in eq 1 to get:

40 - h = 25

h = 40 - 25

h = 15

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The calculated sum of B1,B2,B3 and B4 is 6

From the question, we have the following parameters that can be used in our computation:

BN=(N-1)(N-2)(N-3)

Express properly

So, we have

B(N) = (N-1)(N-2)(N-3)

using the above as a guide, we have the following equations

B(1) = (1-1)(1-2)(1-3) = 0

B(2) = (2-1)(2-2)(2-3) = 0

B(3) = (3-1)(3-2)(3-3) = 0

B(4) = (4-1)(4-2)(4-3) = 6

When the above functions are added, we have

Sum = 0 + 0 + 0 + 6

Evaluate the sum

Sum = 6

Hence, the sum of B1,B2,B3 and B4 is 6

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a. This situation would be a binomial experiment if the following conditions are met:

The trials are independent: The selection of one part does not affect the selection of the other.

Each trial has two possible outcomes: In this case, defective or non-defective.

The probability of success (finding a defective part) remains constant for each trial: In this case, the probability is 3% or 0.03.

The number of trials is fixed: We are selecting two parts, so the number of trials is predetermined.

b. Here is a tree diagram representing the two-trial experiment:

          D                ND

        /   \            /   \

       D     ND         D     ND

The branches represent the two trials, with D representing a defective part and ND representing a non-defective part.

c. To find the number of experimental outcomes resulting in exactly one defect, we can observe that there are two outcomes that meet this criterion: D-ND and ND-D. Both represent one defective part and one non-defective part.

d. The probabilities associated with finding no defects, exactly one defect, and two defects can be calculated as follows:

Probability of finding no defects: (1 - probability of finding a defective part) * (1 - probability of finding a defective part) = (1 - 0.03) * (1 - 0.03) = 0.97 * 0.97 = 0.9409.

Probability of finding exactly one defect: (probability of finding a defective part) * (probability of finding a non-defective part) + (probability of finding a non-defective part) * (probability of finding a defective part) = 0.03 * 0.97 + 0.97 * 0.03 = 0.0582.

Probability of finding two defects: (probability of finding a defective part) * (probability of finding a defective part) = 0.03 * 0.03 = 0.0009.

Therefore, the probabilities associated with finding no defects, exactly one defect, and two defects are approximately 0.9409, 0.0582, and 0.0009, respectively.

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Answer:

To solve the equation 12^(x + 1) = 79 for x, we need to isolate the exponent.

Step 1: Subtract 1 from both sides of the equation:

12^(x + 1) - 1 = 79 - 1

12^(x + 1) - 1 = 78

Step 2: Rewrite 12^(x + 1) as (12^x)(12^1) using the exponent property:

(12^x)(12) - 1 = 78

Step 3: Simplify the left side of the equation:

12(12^x) - 1 = 78

Step 4: Add 1 to both sides of the equation:

12(12^x) = 78 + 1

12(12^x) = 79

Step 5: Divide both sides of the equation by 12:

(12(12^x))/12 = 79/12

12^x = 79/12

Step 6: Take the logarithm (base 12) of both sides of the equation:

log12(12^x) = log12(79/12)

x = log12(79/12)

Therefore, the solution for x is x = log12(79/12)

Hope this helps!

Sin 0 is approximately 0.4709.

To find sin 0 given that cos 0 = 15/17, we can use the Pythagorean identity:

[tex]sin^2 0 + cos^2 0 = 1[/tex]

Rearranging the equation, we have:

[tex]sin^2 0 = 1 - cos^2 0[/tex]

Since we know that cos 0 = 15/17, we can substitute this value into the equation:

[tex]sin^2 0 = 1 - (15/17)^2[/tex]

Calculating this, we find:

[tex]sin^2 0 = 1 - (225/289)\\sin^2 0 = (289/289) - (225/289)\\sin^2 0 = 64/289[/tex]

Taking the square root of both sides, we get:

sin 0 = √(64/289)

Now, we need to determine the sign of sin 0. Since cos 0 is positive (15/17), sin 0 will also be positive in the first and second quadrants.

sin 0 = √(64/289)

sin 0 ≈ √0.2213

sin 0 ≈ 0.4709 (rounded to four decimal places)

Therefore, sin 0 is approximately 0.4709.

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The area of triangle ADE is,

⇒ A = 48 square units

We have to given that,

In the diagram , DE is a midsegment of triangle ABC.

And, The area of triangle ABC is 96 square units

Now, We know that,

Since DE is a midsegment of triangle ABC, it is parallel to AB and half the length of AB. Therefore, DE is half the length of AB.

Hence, the area of triangle ADE is half the area of triangle ABC,

That is,

A = 96 / 2

A = 48 square units

Thus, The area of triangle ADE is,

⇒ A = 48 square units

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We are 95% confident that the true mean time for the execution of an order is between 23.36 and 25.24 minutes.

confidence interval = mean ± 1.96 * standard deviation / ✓(sample size)

In this case, the mean is 24.3 minutes, the standard deviation is 3.2 minutes, and the sample size is 45. Plugging these values into the formula, we get the following confidence interval:

confidence interval = 24.3 ± 1.96 * 3.2 / ✓(45)

= 24.3 ± 0.94

= (23.36, 25.24)

This means that we are 95% confident that the true mean time for the execution of an order is between 23.36 and 25.24 minutes.

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Joel Friedlander is a stockbroker from New York Securities and you are curious about the time that elapses between the placement of a

sales order and its execution. Joel sampled 45 orders and found that the mean time for execution it was 24.3 minutes, with a standard deviation of 3.2 minutes. Help Joel with the construction of a 95% confidence interval for the mean time for the execution of an order

The limit lim(x approaches 0) (cosec(2x) - x*cosec³(x)) is undefined.

We have,

To evaluate the limit of the expression as x approaches 0, we can simplify the expression first.

The expression is given as lim(x approaches 0) (cosec(2x) - x cosec³(x)).

Using trigonometric identities, we can rewrite cosec(2x) as 1/sin(2x) and cosec³(x) as 1/(sin(x))³.

Substituting these into the expression,

We get lim(x approaches 0) (1/sin(2x) - x (1/(sin(x))³)).

Now, let's evaluate the limit term by term:

lim(x approaches 0) (1/sin(2x)) = 1/sin(0) = 1/0 (which is undefined).

lim(x approaches 0) (x (1/(sin(x))³))

= 0 (1/(sin(0))³)

= 0 x 1/0 (which is also undefined).

Since both terms of the expression are undefined as x approaches 0, we cannot determine the limit of the expression.

Therefore,

The limit lim(x approaches 0) (cosec(2x) - x*cosec³(x)) is undefined.

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The correlation coefficient (r) is -0.96.

Given the data:

x: 0, 1, 2, 2, 3, 4, 5, 5, 6

y: 12, 9.5, 9, 8.5, 8.5, 6, 5, 5, 3.5

Step 1: Calculate the mean (average) of x and y.

Mean of x = (0 + 1 + 2 + 2 + 3 + 4 + 5 + 5 + 6) / 9 = 3

Mean of y  = (12 + 9.5 + 9 + 8.5 + 8.5 + 6 + 5 + 5 + 3.5) / 9 = 7

Step 2: Calculate the deviations from the mean for x and y.

Deviation of x (dx) = x - X

Deviation of y (dy) = y - Y

x: -3, -2, -1, -1, 0, 1, 2, 2, 3

y: 5, 2.5, 2, 1.5, 1.5, -1, -2, -2, -3.5

Step 3: Calculate the product of the deviations of x and y.

(dx * dy): -15, -5, -2, -1.5, 0, -1, -4, -4, -10.5

Step 4: Calculate the sum of the products of the deviations (Σ(dx * dy)).

Σ(dx * dy) = -15 - 5 - 2 - 1.5 + 0 - 1 - 4 - 4 - 10.5 = -43

Step 5: Calculate the standard deviations of x (σx) and y (σy).

Standard deviation of x (σx) = √(Σ(dx²) / (n-1))

Standard deviation of y (σy) = √(Σ(dy²) / (n-1))

Calculating dx²:

dx^2: 9, 4, 1, 1, 0, 1, 4, 4, 9

Σ(dx²) = 33

Calculating dy²:

dy²: 25, 6.25, 4, 2.25, 2.25, 1, 4, 4, 12.25

Σ(dy²) = 61

Calculating the standard deviations:

σx = √(33 / (9-1)) = √(33 / 8) ≈ 1.84

σy = √(61 / (9-1)) = √(61 / 8) ≈ 2.70

Step 6: Calculate the correlation coefficient (r).

r = Σ(dx dy) / (√(Σ(dx²)  Σ(dy²)))

r = -43 / (√(33  61))

r ≈ -43 / (√(2013))

r ≈ -43 / 44.87

r ≈ -0.96

Therefore, the correlation coefficient (r) is approximately -0.96.

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The measure of the inscribed angle G in the circle is 85 degree.

An inscribed angle is simply an angle with its vertex on the circle and whose sides are chords.

The relationhip between an an inscribed angle and intercepted arc is expressed as:

Inscribed angle = 1/2 × intercepted arc.

Since the circle equals 360 degrees, we can determine the intercepted arc DF.

Hence:

Arc DF = 360 - ( 70 + 120 )

Arc DF = 360 - 190

Arc DF = 170°

Now, plug the value into the above formula:

Inscribed angle = 1/2 × intercepted arc.

Inscribed angle G = 1/2 × 170°

Inscribed angle G = 85°

Therefore, angle G has a measure of 85 degree.

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Answer:

Please attach a photo to this so I can better answer your question

Answer:

see explanation

Step-by-step explanation:

to find the first 5 terms substitute n = 1, 2, 3, 4, 5 into the explicit formula

36

a₁ = - [tex]3^{1-1}[/tex] = - [tex]3^{0}[/tex] = - 1 [ [tex]a^{0}[/tex] = 1 ]

a₂ = - [tex]3^{2-1}[/tex] = - [tex]3^{1}[/tex] = - 3

a₃ = - [tex]3^{3-1}[/tex] = - 3² = - 9

a₄ = - [tex]3^{4-1}[/tex] = - 3³ = - 27

a₅ = - [tex]3^{5-1}[/tex] = - [tex]3^{4}[/tex] = - 81

the first 5 terms are - 1, - 3, - 9, - 27, - 81

to find a₈ substitute n = 8 into the explicit formula

a₈ = - [tex]3^{8-1}[/tex] = - [tex]3^{7}[/tex] = - 2187

37

to find the first 5 terms substitute n = 1, 2, 3, 4, 5 into the explicit formula

a₁ = 2 × [tex](\frac{1}{2}) ^{1-1}[/tex] = 2 × [tex](\frac{1}{2}) ^{0}[/tex] = 2 × 1 = 2

a₂ = 2 × [tex](\frac{1}{2}) ^{2-1}[/tex] = 2 × [tex](\frac{1}{2}) ^{1}[/tex] = 2 × [tex]\frac{1}{2}[/tex] = 1

a₃ = 2 × [tex](\frac{1}{2}) ^{3-1}[/tex] = 2 × [tex](\frac{1}{2}) ^{2}[/tex] = 2 × [tex]\frac{1}{4}[/tex] = [tex]\frac{1}{2}[/tex]

a₄ = 2 × [tex](\frac{1}{2}) ^{4-1}[/tex] = 2 × ([tex]\frac{1}{2}[/tex] )³ = 2 × [tex]\frac{1}{8}[/tex] = [tex]\frac{1}{4}[/tex]

a₅ = 2 × [tex](\frac{1}{2}) ^{5-1}[/tex] = 2 × [tex](\frac{1}{2}) ^{4}[/tex] = 2 × [tex]\frac{1}{16}[/tex] = [tex]\frac{1}{8}[/tex]

the first 5 terms are 2 , 1 , [tex]\frac{1}{2}[/tex] , [tex]\frac{1}{4}[/tex] , [tex]\frac{1}{8}[/tex]

to find a₈ substitute n = 8 into the explicit formula

a₈ = 2 × [tex](\frac{1}{2}) ^{8-1}[/tex] = 2 × [tex](\frac{1}{2}) ^{7}[/tex] = 2 × [tex]\frac{1}{128}[/tex] = [tex]\frac{1}{64}[/tex]

Answer:

The explicit formula for a geometric sequence is:

a_n = a_1 * r^(n - 1)

where:

In equation 36,

we can see that the first term is -3 and the common ratio is -3. Therefore, we can write the explicit formula for this sequence as:

a_n = -3 * (-3)^(n - 1)

Using this formula, we can find the first five terms and the 8th term in the sequence:

a_1 = -3

a_2 = -3 * (-3) = 9

a_3 = -3 * (-3)^2 = -27

a_4 = -3 * (-3)^3 = 81

a_5 = -3 * (-3)^4 = -243

a_8 = -3 * (-3)^7 = 6561

In equation 37,

we can see that the first term is 2 and the common ratio is 1/2. Therefore, we can write the explicit formula for this sequence as:

a_n = 2 * (1/2)^(n - 1)

Using this formula, we can find the first five terms and the 8th term in the sequence:

a_1 = 2

a_2 = 2 * (1/2) = 1

a_3 = 2 * (1/2)^2 = 1/2= 0.5

a_4 = 2 * (1/2)^3 =1/4= 0.25

a_5 = 2 * (1/2)^4 = 1/8=0.125

a_8 = 2 * (1/2)^7 = 1/64=0.015625

Answer:

To find the points where the rate of change of x is half the rate of change of y, we need to solve the equations:

   dy/dx = k

   dy/dt = k

where k is a constant representing the rate of change.

Step-by-step explanation:

The Equation of circle for Circle B is (x - 2) ² + (y + 4)²  = 38.4.

For circle A:

Radius of circle A = √36 = 6

Area of circle A = π(6)^2 = 36π

Let's assume the radius of circle B is r.

Circumference of circle A = 2π(6) = 12π

Circumference of circle B = 3.2 x Circumference of circle A

= 3.2 x 12π

= 38.4π

Since the ratio of the areas is equal to the ratio of the circumferences, we have:

Area of circle B / Area of circle A = (r^2π) / (36π) = 38.4π / 36π

Simplifying, we get:

r² / 36 = 38.4 / 36

Cross-multiplying, we have:

36 x r² = 38.4 x 36

Dividing both sides by 36:

r^2 = 38.4

Taking the square root of both sides:

r = √38.4

Now we have the radius of circle B. Let's substitute this value into the equation for circle B:

(x - 2) ² + (y + 4)² = (√38.4)²

Simplifying:

(x - 2) ² + (y + 4)²  = 38.4

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The solution to the gaps in the angle proof are:

Multiplication Property of equality

m∠D = ¹/₂ * [arc EFG] and m∠F = ¹/₂ * [arc GDE]

substitution property

The inscribed angle Theorem states that the inscribed angle measures half of the arc of which it is composed.

Thus, we can say that:

m∠D = ¹/₂ * [arc EFG]

and

m∠F = ¹/₂ * [arc GDE]

Therefore:

arc EFG + arc GDE = 360°-------> full circle

Applying multiplication property of equality, we have:

¹/₂ * arc EFG + ¹/₂ * arc GDE = 180°

Applying substitution property of equality, we have:

m∠D = ¹/₂ * [arc EFG]

m∠F = ¹/₂ * [arc GDE]

¹/₂ * arc EFG + ¹/₂ * arc GDE = 180° ----> m∠D + m∠F = 180°

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